Mathematical Physics Vol 1

Chapter 4. Field theory

90

Figure 4.8: Vector line v ( r ) .

As the vector of the tangent t , namely d r , and vector v are collinear, it follows from the definition of a vector line that it’s equation is d r × v = 0 , (4.35) or, on the basis of (1.28), as follows (see Fig. 4.8): d r = v · d t , where t is a parameter . (4.36) In the Cartesian coordinate system (4.36) becomes d x i + d y j + d z k =( v 1 i + v 2 j + v 3 k ) · d t . (4.37) As i , j and k are linearly independent, from the last relation we finally obtain d x v 1 = d y v 2 = d z v 3 = d t . (4.38) This relation represents the differential equation of the vector line. The directional derivative of a vector function is defined similarly to the derivative of a scalar function. d v d s = d v 1 d s i + d v 2 d s j + d v 3 d s k , (4.39) where d v 1 d s = ∂ v 1 ∂ x · d x d s + ∂ v 1 ∂ y · d y d s + ∂ v 1 ∂ z · d z d s ,

∂ v 2

∂ v 2

∂ v 2

d v 2 d s d v 3 d s

d x d s

d y d s

d z d s

(4.40)

∂ x ·

∂ y ·

∂ z ·

=

+

+

,

∂ v 3

∂ v 3

∂ v 3

d x d s

d y d s

d z d s

∂ x ·

∂ y ·

∂ z ·

=

+

+

.

Observe a closed oriented curve in the vector field v , which we will call a contour. A vector line ( l ) passes through each point of this contour.

Definition The geometric locations of vector lines that pass through the points of a contour in a vector field v , form a surface called the solenoid ( tube or vector surface or vector tube ), see Fig. 4.9.

Figure 4.9: Vector surface.